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Thermo-viscous fingering in porous media and in-situ soil remediation Moyles, Iain


The removal of artificially and naturally placed oils found in the ground is of paramount concern for environmental reasons and for the extraction of crude oil for energy. While many companies are involved in this soil remediation process, the physics of the underlying problem are not very well understood. We present a mathematical model and analysis of oil extraction in a porous soil based around the injection of water. We first consider a saturation analysis based on conservation of mass between oil and water in a one-dimensional setting and we find that based on certain parameter values, the water-oil interface goes unstable producing viscous finger patterns. We then include the effects of surface tension between oil and water to determine how this affects the growth of such fingers. We conclude that in the limit of small surface tension effects, the results generalize to the original problem, but more importantly, we deduce that there is a scaling which places the formulation into a setting that is invariant with respect to the magnitude of surface tension effects. With this scaling, we notice that the effect of surface tension is to limit the growth of fingers to a maximal wave number and to prevent their formation entirely beyond a certain critical wave number. Finally with the inclusion of temperature, via heated water injection, we see the formation of a dual fingering pattern: one associated with the mass conservation analysis of the oil-water interface and one associated with the conservation of energy across an interface where thermal gradients occur. We see that the thermal gradients across the interface where temperature drops induce unstable viscous patterns with a higher wave number than would occur for an equivalent isothermal interface where there was solely a change in viscosity. The thermal gradients also promote fingering development downstream across the classical viscosity differential driven interface but at the expense of lowering the interfacial velocity. It is interesting to note that the change in saturation that occurs across the energy interface is a result of a pseudo-free boundary created by the thermal problem.

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